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Theorem plymul0or 22655
Description: Polynomial multiplication has no zero divisors. (Contributed by Mario Carneiro, 26-Jul-2014.)
Assertion
Ref Expression
plymul0or  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( F  oF  x.  G
)  =  0p  <-> 
( F  =  0p  \/  G  =  0p ) ) )

Proof of Theorem plymul0or
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 dgrcl 22608 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
2 dgrcl 22608 . . . . . . 7  |-  ( G  e.  (Poly `  S
)  ->  (deg `  G
)  e.  NN0 )
3 nn0addcl 10838 . . . . . . 7  |-  ( ( (deg `  F )  e.  NN0  /\  (deg `  G )  e.  NN0 )  ->  ( (deg `  F )  +  (deg
`  G ) )  e.  NN0 )
41, 2, 3syl2an 477 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( (deg `  F )  +  (deg
`  G ) )  e.  NN0 )
5 c0ex 9593 . . . . . . 7  |-  0  e.  _V
65fvconst2 6111 . . . . . 6  |-  ( ( (deg `  F )  +  (deg `  G )
)  e.  NN0  ->  ( ( NN0  X.  {
0 } ) `  ( (deg `  F )  +  (deg `  G )
) )  =  0 )
74, 6syl 16 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( NN0  X.  { 0 } ) `  ( (deg
`  F )  +  (deg `  G )
) )  =  0 )
8 fveq2 5856 . . . . . . . 8  |-  ( ( F  oF  x.  G )  =  0p  ->  (coeff `  ( F  oF  x.  G
) )  =  (coeff `  0p ) )
9 coe0 22631 . . . . . . . 8  |-  (coeff ` 
0p )  =  ( NN0  X.  {
0 } )
108, 9syl6eq 2500 . . . . . . 7  |-  ( ( F  oF  x.  G )  =  0p  ->  (coeff `  ( F  oF  x.  G
) )  =  ( NN0  X.  { 0 } ) )
1110fveq1d 5858 . . . . . 6  |-  ( ( F  oF  x.  G )  =  0p  ->  ( (coeff `  ( F  oF  x.  G ) ) `
 ( (deg `  F )  +  (deg
`  G ) ) )  =  ( ( NN0  X.  { 0 } ) `  (
(deg `  F )  +  (deg `  G )
) ) )
1211eqeq1d 2445 . . . . 5  |-  ( ( F  oF  x.  G )  =  0p  ->  ( (
(coeff `  ( F  oF  x.  G
) ) `  (
(deg `  F )  +  (deg `  G )
) )  =  0  <-> 
( ( NN0  X.  { 0 } ) `
 ( (deg `  F )  +  (deg
`  G ) ) )  =  0 ) )
137, 12syl5ibrcom 222 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( F  oF  x.  G
)  =  0p  ->  ( (coeff `  ( F  oF  x.  G ) ) `  ( (deg `  F )  +  (deg `  G )
) )  =  0 ) )
14 eqid 2443 . . . . . . 7  |-  (coeff `  F )  =  (coeff `  F )
15 eqid 2443 . . . . . . 7  |-  (coeff `  G )  =  (coeff `  G )
16 eqid 2443 . . . . . . 7  |-  (deg `  F )  =  (deg
`  F )
17 eqid 2443 . . . . . . 7  |-  (deg `  G )  =  (deg
`  G )
1814, 15, 16, 17coemulhi 22629 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( (coeff `  ( F  oF  x.  G ) ) `
 ( (deg `  F )  +  (deg
`  G ) ) )  =  ( ( (coeff `  F ) `  (deg `  F )
)  x.  ( (coeff `  G ) `  (deg `  G ) ) ) )
1918eqeq1d 2445 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( (
(coeff `  ( F  oF  x.  G
) ) `  (
(deg `  F )  +  (deg `  G )
) )  =  0  <-> 
( ( (coeff `  F ) `  (deg `  F ) )  x.  ( (coeff `  G
) `  (deg `  G
) ) )  =  0 ) )
2014coef3 22607 . . . . . . . 8  |-  ( F  e.  (Poly `  S
)  ->  (coeff `  F
) : NN0 --> CC )
2120adantr 465 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  (coeff `  F
) : NN0 --> CC )
221adantr 465 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  (deg `  F
)  e.  NN0 )
2321, 22ffvelrnd 6017 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( (coeff `  F ) `  (deg `  F ) )  e.  CC )
2415coef3 22607 . . . . . . . 8  |-  ( G  e.  (Poly `  S
)  ->  (coeff `  G
) : NN0 --> CC )
2524adantl 466 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  (coeff `  G
) : NN0 --> CC )
262adantl 466 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  (deg `  G
)  e.  NN0 )
2725, 26ffvelrnd 6017 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( (coeff `  G ) `  (deg `  G ) )  e.  CC )
2823, 27mul0ord 10206 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( (
( (coeff `  F
) `  (deg `  F
) )  x.  (
(coeff `  G ) `  (deg `  G )
) )  =  0  <-> 
( ( (coeff `  F ) `  (deg `  F ) )  =  0  \/  ( (coeff `  G ) `  (deg `  G ) )  =  0 ) ) )
2919, 28bitrd 253 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( (
(coeff `  ( F  oF  x.  G
) ) `  (
(deg `  F )  +  (deg `  G )
) )  =  0  <-> 
( ( (coeff `  F ) `  (deg `  F ) )  =  0  \/  ( (coeff `  G ) `  (deg `  G ) )  =  0 ) ) )
3013, 29sylibd 214 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( F  oF  x.  G
)  =  0p  ->  ( ( (coeff `  F ) `  (deg `  F ) )  =  0  \/  ( (coeff `  G ) `  (deg `  G ) )  =  0 ) ) )
3116, 14dgreq0 22640 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  ( F  =  0p  <->  ( (coeff `  F ) `  (deg `  F ) )  =  0 ) )
3231adantr 465 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( F  =  0p  <->  ( (coeff `  F ) `  (deg `  F ) )  =  0 ) )
3317, 15dgreq0 22640 . . . . 5  |-  ( G  e.  (Poly `  S
)  ->  ( G  =  0p  <->  ( (coeff `  G ) `  (deg `  G ) )  =  0 ) )
3433adantl 466 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( G  =  0p  <->  ( (coeff `  G ) `  (deg `  G ) )  =  0 ) )
3532, 34orbi12d 709 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( F  =  0p  \/  G  =  0p )  <->  ( (
(coeff `  F ) `  (deg `  F )
)  =  0  \/  ( (coeff `  G
) `  (deg `  G
) )  =  0 ) ) )
3630, 35sylibrd 234 . 2  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( F  oF  x.  G
)  =  0p  ->  ( F  =  0p  \/  G  =  0p ) ) )
37 cnex 9576 . . . . . . 7  |-  CC  e.  _V
3837a1i 11 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  CC  e.  _V )
39 plyf 22573 . . . . . . 7  |-  ( G  e.  (Poly `  S
)  ->  G : CC
--> CC )
4039adantl 466 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  G : CC
--> CC )
41 0cnd 9592 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  0  e.  CC )
42 mul02 9761 . . . . . . 7  |-  ( x  e.  CC  ->  (
0  x.  x )  =  0 )
4342adantl 466 . . . . . 6  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  x  e.  CC )  ->  ( 0  x.  x )  =  0 )
4438, 40, 41, 41, 43caofid2 6556 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( CC  X.  { 0 } )  oF  x.  G )  =  ( CC  X.  { 0 } ) )
45 id 22 . . . . . . . 8  |-  ( F  =  0p  ->  F  =  0p
)
46 df-0p 22055 . . . . . . . 8  |-  0p  =  ( CC  X.  { 0 } )
4745, 46syl6eq 2500 . . . . . . 7  |-  ( F  =  0p  ->  F  =  ( CC  X.  { 0 } ) )
4847oveq1d 6296 . . . . . 6  |-  ( F  =  0p  -> 
( F  oF  x.  G )  =  ( ( CC  X.  { 0 } )  oF  x.  G
) )
4948eqeq1d 2445 . . . . 5  |-  ( F  =  0p  -> 
( ( F  oF  x.  G )  =  ( CC  X.  { 0 } )  <-> 
( ( CC  X.  { 0 } )  oF  x.  G
)  =  ( CC 
X.  { 0 } ) ) )
5044, 49syl5ibrcom 222 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( F  =  0p  -> 
( F  oF  x.  G )  =  ( CC  X.  {
0 } ) ) )
51 plyf 22573 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  F : CC
--> CC )
5251adantr 465 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  F : CC
--> CC )
53 mul01 9762 . . . . . . 7  |-  ( x  e.  CC  ->  (
x  x.  0 )  =  0 )
5453adantl 466 . . . . . 6  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  /\  x  e.  CC )  ->  ( x  x.  0 )  =  0 )
5538, 52, 41, 41, 54caofid1 6555 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( F  oF  x.  ( CC  X.  { 0 } ) )  =  ( CC  X.  { 0 } ) )
56 id 22 . . . . . . . 8  |-  ( G  =  0p  ->  G  =  0p
)
5756, 46syl6eq 2500 . . . . . . 7  |-  ( G  =  0p  ->  G  =  ( CC  X.  { 0 } ) )
5857oveq2d 6297 . . . . . 6  |-  ( G  =  0p  -> 
( F  oF  x.  G )  =  ( F  oF  x.  ( CC  X.  { 0 } ) ) )
5958eqeq1d 2445 . . . . 5  |-  ( G  =  0p  -> 
( ( F  oF  x.  G )  =  ( CC  X.  { 0 } )  <-> 
( F  oF  x.  ( CC  X.  { 0 } ) )  =  ( CC 
X.  { 0 } ) ) )
6055, 59syl5ibrcom 222 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( G  =  0p  -> 
( F  oF  x.  G )  =  ( CC  X.  {
0 } ) ) )
6150, 60jaod 380 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( F  =  0p  \/  G  =  0p )  ->  ( F  oF  x.  G
)  =  ( CC 
X.  { 0 } ) ) )
6246eqeq2i 2461 . . 3  |-  ( ( F  oF  x.  G )  =  0p  <->  ( F  oF  x.  G )  =  ( CC  X.  { 0 } ) )
6361, 62syl6ibr 227 . 2  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( F  =  0p  \/  G  =  0p )  ->  ( F  oF  x.  G
)  =  0p ) )
6436, 63impbid 191 1  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( ( F  oF  x.  G
)  =  0p  <-> 
( F  =  0p  \/  G  =  0p ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1383    e. wcel 1804   _Vcvv 3095   {csn 4014    X. cxp 4987   -->wf 5574   ` cfv 5578  (class class class)co 6281    oFcof 6523   CCcc 9493   0cc0 9495    + caddc 9498    x. cmul 9500   NN0cn0 10802   0pc0p 22054  Polycply 22559  coeffccoe 22561  degcdgr 22562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573  ax-addf 9574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-fal 1389  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-of 6525  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-map 7424  df-pm 7425  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-sup 7903  df-oi 7938  df-card 8323  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10214  df-nn 10544  df-2 10601  df-3 10602  df-n0 10803  df-z 10872  df-uz 11093  df-rp 11232  df-fz 11684  df-fzo 11807  df-fl 11911  df-seq 12090  df-exp 12149  df-hash 12388  df-cj 12914  df-re 12915  df-im 12916  df-sqrt 13050  df-abs 13051  df-clim 13293  df-rlim 13294  df-sum 13491  df-0p 22055  df-ply 22563  df-coe 22565  df-dgr 22566
This theorem is referenced by:  plydiveu  22672  quotcan  22683  vieta1lem1  22684  vieta1lem2  22685
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